scholarly journals Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Jian Deng
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Stability ◽  
Random Fluctuation ◽  
Largest Lyapunov Exponent ◽  
Real Noise ◽  
The Largest Lyapunov Exponent ◽  
The Stability ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment stochastic stability and almost-sure stochastic stability of two-degree-of-freedom coupled viscoelastic systems, under the parametric excitation of a real noise, are investigated through the moment Lyapunov exponents and the largest Lyapunov exponent, respectively. The real noise is also called the Ornstein-Uhlenbeck stochastic process. For small damping and weak random fluctuation, the moment Lyapunov exponents are determined approximately by using the method of stochastic averaging and a formulated eigenvalue problem. The largest Lyapunov exponent is calculated through its relation with moment Lyapunov exponents. The stability index, the stability boundaries, and the critical excitation are obtained analytically. The effects of various parameters on the stochastic stability of the system are then discussed in detail. Monte Carlo simulation is carried out to verify the approximate results of moment Lyapunov exponents. As an application example, the stochastic stability of a flexural-torsional viscoelastic beam is studied.

Lyapunov Exponents and Stochastic Stability of Two-Dimensional Parametrically Excited Random Systems

1993 ◽  
Vol 60 (3) ◽  
pp. 677-682 ◽  
Author(s):  
S. T. Ariaratnam ◽  
Wei-Chau Xie
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Random Systems ◽  
Random Fluctuation ◽  
Largest Lyapunov Exponent ◽  
Random Disturbance ◽  
Two Dimensional ◽  
Parametrically Excited ◽  
The Largest Lyapunov Exponent

The variation of the largest Lyapunov exponent for two-dimensional parametrically excited stochastic systems is studied by a method of linear transformation. The sensitivity to random disturbance of systems undergoing bifurcation is investigated. Two commonly occurring examples in structural dynamics are considered, where the random fluctuation appears in the stiffness term or the damping term. The boundaries of almost-sure stochastic stability are determined by the vanishing of the largest Lyapunov exponent of the linearized system. The validity of the approximate results is checked by numerical simulation.

Moment Lyapunov Exponents of a Two-Dimensional Viscoelastic System Under Bounded Noise Excitation

2002 ◽  
Vol 69 (3) ◽  
pp. 346-357 ◽  
Author(s):  
W.-C. Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Regular Perturbation ◽  
Stochastic Fluctuation ◽  
Viscoelastic System ◽  
Moment Lyapunov Exponent ◽  
The Moment ◽  
Moment Lyapunov Exponents

The moment Lyapunov exponents of a two-dimensional viscoelastic system under bounded noise excitation are studied in this paper. An example of this system is the transverse vibration of a viscoelastic column under the excitation of stochastic axial compressive load. The stochastic parametric excitation is modeled as a bounded noise process, which is a realistic model of stochastic fluctuation in engineering applications. The moment Lyapunov exponent of the system is given by the eigenvalue of an eigenvalue problem. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter. The results obtained are compared with those for which the effect of viscoelasticity is not considered.

scholarly journals MOMENT LYAPUNOV EXPONENTS AND STOCHASTIC STABILITY OF A THIN-WALLED BEAM SUBJECTED TO AXIAL LOADS AND END MOMENTS

2021 ◽  
Vol 19 (2) ◽  
pp. 209
Author(s):  
Goran Janevski ◽  
Predrag Kozić ◽  
Ratko Pavlović ◽  
Strain Posavljak
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Degrees Of Freedom ◽  
Simulation Method ◽  
Stochastic Dynamic ◽  
Regular Perturbation ◽  
Thin Walled ◽  
The Moment ◽  
Moment Stability ◽  
Moment Lyapunov Exponents

In this paper, the Lyapunov exponent and moment Lyapunov exponents of two degrees-of-freedom linear systems subjected to white noise parametric excitation are investigated. The method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises. The Lyapunov exponent and moment Lyapunov exponents are important characteristics for determining both the almost-sure and the moment stability of a stochastic dynamic system. As an example, we study the almost-sure and moment stability of a thin-walled beam subjected to stochastic axial load and stochastically fluctuating end moments.  The validity of the approximate results for moment Lyapunov exponents is checked by numerical Monte Carlo simulation method for this stochastic system.

Weak Noise Expansion of Moment Lyapunov Exponents of a Two-Dimensional System Under Bounded Noise Excitation

2000 ◽  
Author(s):  
Wei-Chau Xie
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Dimensional System ◽  
Two Dimensional ◽  
Bounded Noise ◽  
Regular Perturbation ◽  
Weak Noise ◽  
Two Dimensional System ◽  
The Moment ◽  
Moment Lyapunov Exponents

Abstract The moment Lyapunov exponents of a two-dimensional system under bounded noise parametric excitation are studied in this paper. The method of regular perturbation is applied to obtain weak noise expansions of the moment Lyapunov exponent, Lyapunov exponent, and stability index in terms of the small fluctuation parameter.

Stochastic Stability of Gyroscopic Systems Under Bounded Noise Excitation

2018 ◽  
Vol 18 (02) ◽  
pp. 1850022 ◽  
Author(s):  
Jian Deng
Keyword(s):  
Lyapunov Exponent ◽  
Parametric Resonance ◽  
Stochastic Stability ◽  
Largest Lyapunov Exponent ◽  
Gyroscopic System ◽  
Bounded Noise ◽  
Rotating Shaft ◽  
Dynamic Stochastic ◽  
The Largest Lyapunov Exponent ◽  
Gyroscopic Systems

Dynamic stochastic stability of a two-degree-of-freedom gyroscopic system under bounded noise parametric excitation is studied in this paper through moment Lyapunov exponent and the largest Lyapunov exponent. A rotating shaft subject to stochastically fluctuating thrust is taken as a typical example. To obtain these two exponents, the gyroscopic differential equation of motion is first decoupled into Itô stochastic differential equations by using the method of stochastic averaging. Then mathematical transformations are used in these Itô equation to obtain a partial differential eigenvalue problem governing moment Lyapunov exponents, the slope of which at the origin is equal to the largest Lyapunov exponent. Depending upon the numerical relationship between the natural frequency and the excitation frequencies, the gyroscopic system may fall into four types of parametric resonance, i.e. no resonance, subharmonic resonance, combination additive resonance, and combination differential resonance. The effects of noise and frequency detuning parameters on the parametric resonance are investigated. The results pave the way to utilize or control the vibration of gyroscopic systems under stochastic excitation.

ON THE LOCAL STOCHASTIC STABILITY OF NONLINEAR COMPLEX NETWORKS

2010 ◽  
Vol 20 (01) ◽  
pp. 177-184 ◽  
Author(s):  
ZHI-LONG HUANG ◽  
ZHOU YAN ◽  
XIAO-LING JIN ◽  
GUANRONG CHEN
Keyword(s):  
Lyapunov Exponent ◽  
Theoretical Analysis ◽  
Complex Networks ◽  
Stochastic Stability ◽  
Largest Lyapunov Exponent ◽  
Stochastic Perturbations ◽  
Approximate Analytical Solutions ◽  
Coupling Strengths ◽  
The Largest Lyapunov Exponent ◽  
The One

The local stochastic stability of nonlinear complex networks is studied, subject to stochastic perturbations to the coupling strengths and stochastic parametric excitations to the nodes. The complex network is first linearized at its trivial solution and then the linearized network is reduced to N independent subsystems by using a suitable linear transformation, where N is the size of the network. The largest Lyapunov exponent for each subsystem is then calculated and all the approximate analytical solutions are evaluated for some specific cases. It is found that the largest Lyapunov exponent among all subsystems is the one associated with the subsystem that has the largest or the smallest eigenvalue of the configuration matrix of the network. Finally, an example is given to demonstrate the validity and accuracy of the theoretical analysis.

scholarly journals Alternative Methods of the Largest Lyapunov Exponent Estimation with Applications to the Stability Analyses Based on the Dynamical Maps—Introduction to the Method

Materials ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 7197
Author(s):  
Artur Dabrowski ◽  
Tomasz Sagan ◽  
Volodymyr Denysenko ◽  
Marek Balcerzak ◽  
Sandra Zarychta ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Numerical Algorithms ◽  
Alternative Methods ◽  
Largest Lyapunov Exponent ◽  
Continuous Systems ◽  
Science And Engineering ◽  
Stability Of Dynamical Systems ◽  
The Largest Lyapunov Exponent ◽  
The Stability ◽  
Mathematical Operations

Controlling stability of dynamical systems is one of the most important challenges in science and engineering. Hence, there appears to be continuous need to study and develop numerical algorithms of control methods. One of the most frequently applied invariants characterizing systems’ stability are Lyapunov exponents (LE). When information about the stability of a system is demanded, it can be determined based on the value of the largest Lyapunov exponent (LLE). Recently, we have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. The present article introduces new methods of LLE estimation for continuous systems and maps. We have shown that application of our approaches will introduce significant improvement of the efficiency. We have also proved that our approach is simpler and more efficient than commonly applied algorithms. Moreover, as our approach works in the case of dynamical maps, it also enables an easy application of this method in noncontinuous systems. We show comparisons of efficiencies of algorithms based our approach. In the last paragraph, we discuss a possibility of the estimation of LLE from maps and for noncontinuous systems and present results of our initial investigations.

Moment Lyapunov Exponent and Stochastic Stability of Two Coupled Oscillators Driven by Real Noise

2001 ◽  
Vol 68 (6) ◽  
pp. 903-914 ◽  
Author(s):  
N. Sri Namachchivaya ◽  
H. J. Van Roessel
Keyword(s):  
Lyapunov Exponent ◽  
Asymptotic Approximation ◽  
Coupled Oscillators ◽  
Stability Index ◽  
Stationary Solutions ◽  
Real Noise ◽  
Moment Lyapunov Exponent ◽  
The Stability ◽  
The Moment

In a recent paper an asymptotic approximation for the moment Lyapunov exponent, gp, of two coupled oscillators driven by a small intensity real noise was obtained. The utility of that result is limited by the fact that it was obtained under the assumption that the moment p is small, a limitation which precludes, for example, the determination of the stability index. In this paper that limitation is removed and an asymptotic approximation valid for arbitrary p is obtained. The results are applied to study the moment stability of the stationary solutions of structural and mechanical systems subjected to stochastic excitation.

scholarly journals Largest Lyapunov Exponents and Bifurcations of Stochastic Nonlinear Systems

1996 ◽  
Vol 3 (4) ◽  
pp. 313-320 ◽  
Author(s):  
C.W.S. To ◽  
D.M. Li
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Systems ◽  
Random Excitation ◽  
Van Der Pol Oscillator ◽  
Stochastic Nonlinear Systems ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Points ◽  
Largest Lyapunov Exponents ◽  
The Largest Lyapunov Exponent

Two commonly adopted expressions for the largest Lyapunov exponents of linearized stochastic systems are reviewed. Their features are discussed in light of bifurcation analysis and one expression is selected for evaluating the largest Lyapunov exponent of a linearized system. An independent method, developed earlier by the authors, is also applied to determine the bifurcation points of a van der Pol oscillator under parametric random excitation. It is shown that the bifurcation points obtained by the independent technique agree qualitatively and quantitatively with those evaluated by using the largest Lyapunov exponent of the linearized oscillator.

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